To formulate \(\Pi \)  within underdamped dynamics, we explore the link
between the entropy production rate and the heat dissipation rate [1]}, [2]}, [3]}, [4]}, [5]}.
Using Kramers equation, we obtain
a number of alternative formulations of \(\Pi \) . One by-product
of this analysis is the derivation of the virial theorem for active particles. Another interesting result is the
representation of distributions in \(v\) -space, for active particles in unconfined environment,
as convolution between the Maxwell distribution and the distribution at zero temperature.
